If limx→∞ 1+ax−4x22x=e3, then ‘a’ is equal to

# If $\underset{x\to \mathrm{\infty }}{lim} {\left(1+\frac{a}{x}-\frac{4}{{x}^{2}}\right)}^{2x}={e}^{3},$ then $\text{'}a\text{'}$ is equal to

1. A

$\frac{2}{3}$

2. B

$\frac{3}{2}$

3. C

2

4. D

$\frac{1}{2}$

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### Solution:

We have,

$\begin{array}{l}\underset{x\to \mathrm{\infty }}{lim} {\left(1+\frac{a}{x}-\frac{4}{{x}^{2}}\right)}^{2x}={e}^{3}\\ ⇒{e}^{\underset{x\to \infty }{lim} 2x\left(\frac{a}{x}-\frac{4}{{x}^{2}}\right)}={e}^{3}\\ ⇒\underset{x\to \infty }{lim} \left(2a-\frac{8}{x}\right)=3⇒2a=3⇒a=\frac{3}{2}\end{array}$

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